Ample vector bundles with sections vanishing on submanifolds of sectional genus three

Let E be an ample vector bundle of rank r geq 2 on a smooth complex projective variety X of dimension n such that there exists a global section of E whose zero locus Z is a smooth subvariety of dimension n-r geq 2 of X. Let H be an ample line bundle on X such that |H| defines an embedding of Z. The triplets (X,E,H) such that (Z,H_Z) has sectional genus g(Z,H_Z) = 3 are classified. This is the first step towards the classification of ample vector bundles of rank n-1 on X of curve genus three.